1. Measuring Entanglement Entropy in a Many-body System
K. Rajibul Islam
MIT-Harvard Center for Ultracold Atoms
Caltech
Mar 08, 2016
HARVARD UNIVERSITY  MIT
CENTER FOR ULTRACOLD ATOMS

2. Strongly interacting quantum systems
Wikipedia.org
High Temperature superconductor
Quark Gluon ‘plasma’
Microscopic
description?
Spin network
Interacting atoms
Macroscopic
phenomenon?
Simulating quantum matter on computers?

3. Quantum Superposition
Exponential growth of Hilbert space
Exponential growth of Hilbert space
For N qubits - No. of states = 2N
N = 40
~ 1 Tb
Entanglement
Growth of Entanglement – hard to compute
Solving Quantum dynamics of interacting spin models currently limited to about spins.

4. Quantum Simulation 2S1/2 = |1,0 Spin states can be initialized and
Feynman, International Journal of Theoretical Physics, Vol 21, No. 6/7, 1982
Lloyd, Science, Vol 273, No. 5278, 1996
2S1/2
= |0,0
= |1,0
qubits
Spin states can be initialized and
Individually detected
Long coherence – up to 15 minutes!

5. Quantum Simulation : Platforms
Trapped ions
Nature Physics 8, 277–284 (2012) 
Polar molecules
Nature 501, 521 (2013)
Neutral atoms in optical lattices
Nature Physics 8, 
267–276 (2012) 
Photonic networks
Nature Physics 8, 
285–291 (2012)
Superconducting circuits
Nature Physics 8, 292–299 (2012) 
Defects in diamonds
Physics Today 67(10), 38(2014)

6. ? Quantum Simulation : Trapped Ions Ising + ‘Bottom-up’ approach
Tunable interactions – quantum Ising, XY, XYZ … ?
Frustrated!!
Ising +
Entanglement in the ground state
K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin, L.-M. Duan, C. Monroe Nature 465, 590 (2010).

7. Quantum Simulation : Trapped Ions
‘Quantum phase transitions’
𝐽 𝑖,𝑗 ≈ 𝐽 0 /|𝑖−𝑗|
N = 10
R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J. K. Freericks, and C. Monroe Nature Communications 2:377 (2011)
R. Islam, C. Senko, W. C. Campbell, S. Korenblit, J. Smith, A. Lee, E. E. Edwards, C.-C. J. Wang, J. K. Freericks, and C. Monroe, Science 340, 583 (2013).

8. Superconducting circuits
Quantum Simulation : Platforms
Trapped ions
Nature Physics 8, 277–284 (2012) 
Neutral atoms
in optical lattices
Nature Physics 8, 
267–276 (2012) 
Superconducting circuits
Nature Physics 8, 
292–299 (2012) 
NV defects in diamonds
Physics Today 67(10), 38(2014)
Photonic
networks
Nature Physics 
8, 285–291 (2012)

9. Quantum gas microscope
Bakr et al., Nature 462, 74 (2009), Bakr et al., Science (June 2010)
Previous work on single site addressability in lattices: Detecting single atoms in large spacing lattices (D. Weiss) and 1D standing waves (D. Meschede), Electron Microscope (H. Ott), Absorption imaging (J. Steinhauer), single trap (P. Grangier, Weinfurter/Weber), few site resolution (C. Chin), See also: Sherson et al., Nature 467, 68 (2010) 9

10. Single site parity Imaging

11. Quantum gas microscope
High aperture objective NA=0.8
High resolution imaging
2D quantum gas of Rb-87
in optical lattice
Hologram for projecting optical lattice 11

12. Bose Hubbard Model U/J tunneling J interaction U Superfluid
Mott insulator
Superfluid
U/J
Bakr et al., Science. 329, 547 (2010)

13. Projecting arbitrary potential landscapes
hologram
Fourier hologram
Image: EKB Technologies
objective
2D quantum gas of Rb-87
in optical lattice
Thesis : P. Zupancic (LMU/Harvard, 2014) 13

14. Arbitrary beam shaping
Weitenberg et al., Nature 471, (2011) Zupancic, P., Master’s Thesis, LMU Munich/Harvard 2013
Cizmar, T et al., Nature Photonics 4, 6 (2010)
High-order Laguerre Modes
Laguerre-Gauss profile 1
10-1
10-2
10-3

15. A bottom-up system for neutral atoms
(Single shot image)

16. Single-Particle Bloch oscillationsF
P. M. Preiss, R. Ma, M. E. Tai, A. Lukin, M. Rispoli, P. Zupancic, Y. Lahini, R. Islam, M. Greiner Science 347, 1229 (2015)

17. Single-Particle Bloch oscillationsF
Temporal period , spatial width
Delocalized over ~14 sites = 10μm.
Revival probability 96(3)%
P. M. Preiss, R. Ma, M. E. Tai, A. Lukin, M. Rispoli, P. Zupancic, Y. Lahini, R. Islam, M. Greiner Science 347, 1229 (2015)

18. Entanglement in Many-body Systems
Novel states of matter:
Order beyond simple broken symmetry
Example - Topological order, spin liquid, fractional quantum Hall - characterized by quantum entanglement !
Quantum criticality
Quantum dynamics …
Challenge: Entanglement not detected in traditional CM experiments
Entanglement in ultra-cold atom synthetic quantum matter?

19. Entanglement in Many-body Systems
A B
Many-body system: Bipartite entanglement
Product state: e.g. Mott insulator
Entangled state: e.g. Superfluid

20. Entanglement Entropy TRACE A B Tr( 𝜌 𝐴 2 ) =1 <1
Reduced density matrix:
Product state
 Pure state
Entangled state
 Mixed state
Tr( 𝜌 𝐴 2 )
Quantum purity = =1
<1
𝑆 2 𝜌 𝐴 =− log Tr( 𝜌 𝐴 2 )
Renyi Entanglement Entropy

21. Entanglement Entropy TRACE A B Tr( 𝜌 𝐴 2 ) =1 <1
Reduced density matrix:
Product state
 Pure state
Entangled state
 Mixed state
Tr( 𝜌 𝐴 2 )
Quantum purity = =1
<1
Many-body Hong-Ou-Mandel interferometry
Alves and Jaksch, PRL 93, (2004)
Mintert et al., PRL 95, (2005)
Daley et al., PRL 109, (2012)

22. No coincidence detection
Hong-Ou-Mandel interference
No coincidence detection
for identical photons
Hong C. K., Ou Z. Y., and Mandel L. Phys. Rev. Lett (1987)

23. Beam splitter operation: Rabi flopping in a double well
P (R) L R -i +i
Also see: Kaufman A M et al., Science 345, 306 (2014)
Without single atom detection:
Trotzky et al., PRL 105, (2010)
also Esslinger group

24. Two bosons on a beam splitter
Hong-Ou-Mandel interference
Beam splitter

25. limited by interaction measured fidelity:
Beam splitter
Beam splitter
P(1,1)
4(4)%
96(4)%
limited by interaction
measured fidelity:
Time in double well (ms)
Also see : Kaufman A M et al., Science 345, 306 (2014),
R. Lopes et al, Nature 520, 7545 (2015)

26. Quantum interference of bosonic many body systems?
How “identical” are the particles?
vs.
How “identical” are the states?
If , deterministic number parity after beam splitter
Alves and Jaksch, PRL 93 (2004)
Daley et al., PRL 109 (2012)

27. Quantum interference of bosonic many body systems

28. Making two copies of a many-body state

29. Measuring many-body entanglement
Mott Insulator
Locally
pure
Product State
Globally pure
Superfluid
Locally
mixed
Entangled
Globally pure

30. Measuring many-body entanglement
Mott Insulator
always even
 locally pure
HOM
even even  globally pure
Superfluid
odd or even
 locally mixed
HOM
 Entangled!
even even  globally pure
Ref: Alves C M, Jaksch D, PRL 93, (2004), Daley A J et al, PRL 109, (2012)

31. Entanglement in the ground state of a Bose-Hubbard system
Mixed H
Purity = Parity
Renyi entropy
Purity
Beam splitter
complete
2-site
1-site
Pure
Rajibul Islam et al,
Nature 528, 77 (2015)
U/J
Mott
insulator
Superfluid
Entanglement in optical lattice systems:
M. Cramer et al, Nature Comm, 4 (2013),
T. Fukuhara et al, PRL 115, (2015)

32. Entanglement in the ground state of a Bose-Hubbard system
Mixed H
Purity = Parity
Renyi entropy
Purity
Beam splitter
complete
2-site
1-site
Pure
Rajibul Islam et al,
Nature 528, 77 (2015)
U/J
Mott
insulator
Superfluid

33. Entanglement in the ground state of a Bose-Hubbard system
Mixed H
Purity = Parity
Renyi entropy
Beam splitter
complete
2-site
1-site
Pure
Rajibul Islam et al,
Nature 528, 77 (2015)
U/J
Mott
insulator
Superfluid

34. Entanglement in the ground state of a Bose-Hubbard system
Mixed H
Purity = Parity
Renyi entropy
Purity
Beam splitter
complete
2-site
1-site
Pure
Rajibul Islam et al,
Nature 528, 77 (2015)
U/J
Mott
insulator
Superfluid

35. Entanglement in the ground state of a Bose-Hubbard system
Renyi entropy
complete
2-site
1-site
U/J
Mott
insulator
Superfluid

36. Mutual Information IAB
IAB = S2(A) + S2(B) - S2(AB)
Renyi entropy
IAB
Mutual Information
Boundary Area
U/J
Mott
insulator
Superfluid

37. Non equilibrium: Quench dynamics
Beam splitter

38. Non equilibrium- Quench dynamics
Outlook:
Non equilibrium- Quench dynamics
Greiner group - unpublished

39. Scaling of entanglement entropy and mutual information – probe critical points, violation of area law etc.
Dynamical phenomena with entanglement – MBL phase.
Overlap of two wave functions
Sensitivity to perturbation signaling quantum phase transitions.
Higher order Renyi entropies by interfering more than two copies. Ψ1 Ψ2

40. A ‘quantum gas microscope’ for ions

41. Thank you!! Theory Experiments Harvard Philipp Preiss Ruichao Ma
Eric Tai
Matthew Rispoli
Alex Lukin
Markus Greiner
Maryland
Kihwan Kim (Now at Tshinhua, China)
Ming-Shien Chang (now at Academia Sinica, Taiwan)
Emily Edwards
Wes Campbell (now at UCLA)
Simcha Korenblit (now at Hebrew University)
Crystal Senko (now at Harvard)
Jake Smith
Aaron Lee
Chris Monroe
Guin-Dar Lin (Michigan)
Luming Duan (Michigan)
Joseph C.-C. Wang
(Georgetown)
Jim Freericks (Georgetown)
Changsuk Noh (Auckland)
Howard Carmichael (Auckland)
Andrew Daley (strathclyde)
Peter Zoller (Innsbruck)
Eugene Demler (Harvard)